AUC Optimization for Multiclass AUC Optimization. There are few studies that focus on AUC optimization algorithms for multiclass problems, which fall in the following two directions. The first direction of studies focuses on the multipartite ranking problem, which is a natural extension of the bipartite ranking problem where the order is presented with more than two discrete degrees. Hitherto, a substantial amount of efforts have been made to explore the AUC optimization method/theory for the multipartite ranking problem [1]}, [2]}, [3]}, [4]}, [5]}. Moreover, a recent work [6]} proposes a novel nonlinear semi-supervised multipartite ranking problem for large-scale datasets. It is noteworthy that multipartite ranking approaches could solve multiclass problems only if the classes are ordinal values for the same semantic concept. For example, the age estimation task could be regarded as a multiclass problem where the class labels are the ages for a given person; the movie rating prediction could be regarded as a multiclass problem where the class labels are the ratings for the same given movie. The major difference here is that we focus on the generic multiclass problems where different labels present different semantic concepts and do not have clear ordinal relations. Thus the studies along this line are not available to our setting in general. Most recently, [7]} discusses the possibility of exploring AUC optimization for general multiclass settings. The major differences are as follows. First, [7]} only focuses on the square loss function, while our study presents a general framework for multiclass AUC optimization. Second, [7]} focuses more on the optimization properties, where the original problem is reformulated as a minimax problem. By contrast, our study focuses on the learning properties for multiclass AUC optimization, such as its generalization ability and the consistency of different loss functions. Moreover, we also present a series of acceleration methods that do not require any reformulation of the optimization problem.
